Abstract
In this research article, we presented the idea of intuitionistic fuzzy incidence graphs (IFIGs) along with their certain properties. The number of operations including Cartesian product (CP), composition, tensor product, and normal product in an IFIGs are also investigated. The method to compute the degree of IFIGs obtained by CP, composition, tensor product, and the normal product is discussed. Some important theorems to calculate the degree of the vertices of IFIGs acquired by CP, composition, tensor product, and normal product are elaborated. An application of CP and composition of two IFIGs in the textile industry to find the best combinations of departments expressing the highest percentage of progress and the lowest percentage of nonprogress is provided. A comparative analysis of our study with the existing study is discussed. Our study will be beneficial to comprehend and understand the further characteristics of IFIGs in detail. Another advantage of our study is that it will be helpful to find the maximum percentage of progress and minimum percentage of nonprogress in different departments of universities, garment factories, and hospitals.
1. Introduction
Zadeh [1] initiated the idea of a fuzzy set (FS) to resolve difficulties in dealing with uncertainties. Since then, the theory of FSs and fuzzy logic have been examined by many researchers to solve many real-life problems involving an ambiguous and uncertain environment. FS theory becomes a sturdy area in multiple disciplines including mathematics, computer science, and signal processing. FSs were not without flaws, they only talked about the membership function (MSF) and missed the nonmembership function (NMSF). This drawback in FSs leads Attnassov [2] to present the idea of the intuitionistic fuzzy set (IFS) as a generalization of FS that handles unsure situations in a good way as its structure is not restricted to membership (MS) grades only. The concept of IFS is a preferable tool to use due to its sundry structure explaining MS as well as nonmembership (NMS) grades of an element. IFSs have been outstandingly used in different areas of life. De et al. [3] presented some applications of IFSs in medical diagnosis. Xu [4] defined some aggregation operators.
A graph is an easy way of expressing information including an association between entities. The entities are shown by vertices and relations by edges. In different problems, we get incomplete information about the problem. So, there is blurriness in the explanation of the entities or their relationship or both. To tackle this form of problem, we need to design a fuzzy graph (FG) model. Zadeh’s FS provided a strong ground for the theory of FGs which have been proposed by Rosenfeld [5]. The study of FGs leads many researchers to contribute in this field; for example, Mordeson and Chang-Shyh [6] explained the various kinds of operations like composition, union, and join on FGs. Pal et al. [7] established the degree and total degree of an edge in interval-valued FGs. Some properties of highly irregular interval-valued FGs were initiated by Rashmanlou and Pal [8]. Jan et al. [9] discussed some root-level modifications in interval-valued FGs. Mathew et al. [10] introduced vertex rough graphs. Certain types of FGs such as fuzzy threshold graphs, fuzzy tolerance graphs, and Bipolar fuzzy hypergraphs were given by Samanta and Pal [11–13]. For more comprehensive and detailed work on graphs and FGs, we may refer to the reader [14–16].
Parvathi and Karunambigai [17] have developed the idea of intuitionistic fuzzy graphs (IFGs). Like IFSs, some outstanding work on the theory of IFGs is also being done. Parvathi et al. [18] discussed different operations on IFGs including CP and composition. Gani and Begum [19] talked about the degree, order, and size of IFGs. Direct product, semistrong product, and strong product in IFGs were discussed by Sahoo and Pal [20]. Sahoo and Pal [21, 22] studied intuitionistic fuzzy competition graphs and intuitionistic fuzzy tolerance graphs with applications. Various kinds of products like Cartesian, tensor, normal, and their degrees in IFGs were provided by Sahoo and Pal [23]. Sahoo et al. [24] initiated new ideas in IFGs with application in the water supply system. Bozhenyuk et al. [25] presented an idea of minimal intuitionistic dominating vertex subset of an IFG.
FGs and IFGs do not talk about the influence of vertices on the edges. This lack in these graphs was caused to introduce an idea of fuzzy incidence graphs (FIGs). For example, if vertices show different residence societies and edges show roads joining these residence societies, we can have a FG expressing the extent of traffic from one society to another. The society that has the maximum number of residents will have maximum ramps in society. So, if c and d are two societies and cd is a road joining them then (c, cd) could express the ramp system from the road cd to the society c. In the case of an unweighted graph, both c and d will have an influence of 1 on cd. In a directed graph, the influence of c on cd represented by (c, cd) is 1 whereas (d, cd) is 0. This concept is generalized through FIGs. Dinesh [26] presented the notion of FIGs. Malik et al. [27, 28] utilized FIGs in human trafficking. The idea of cut-pairs, fuzzy incidence trees, and strong pairs in FIGs was proposed by Mathew and Mordeson [29]. Fang et al. [30] presented an idea to find the connectivity and Wiener index of FIG.
IFGs and different types of operations for IFGs including CP, composition, tensor product (TP), and normal product (NP) exist in literature but these operations are unknown for IFIGs. This motivates us to introduce IFIGs and these operations for IFIGs. IFIGs are more beneficial than FIGs due to the availability of the NMSF in IFIGs. These innovative ideas will open a new door for many researchers to study IFIGs in detail. The remainder of this article is formulated as follows: Section 2 provides some preliminary results which are required to understand the remaining part of the article. The method to find the degree of a vertex in CP and composition is discussed in Section 3. In Section 4, the idea to find the degree of a vertex in TP and NP is explained. The idea of accurate domination in FIGs is presented in Section 5. A real-life application of CP and composition in the textile industry is provided in Section 5. A comparative analysis of our study with the existing study is provided in Section 6. Conclusions and prospects are elaborated in Section 7.
2. Preliminaries
A fuzzy subset (FSS) on a set is a map . A map is known as a fuzzy relation on if for each . A is a pair , where is a FSS on a set and is a fuzzy relation on .
Definition 1 (see [23]). An IFS on the set is characterized by a mapping , which is named as a MF and , which is said to be NMF.
Definition 2 (see [23]). An IFG is of the form where , , and such that and represent the degree of MS and NMS of the vertex , respectively, and for each . Also, and ; and show the degree of MS and NMS of the edge , respectively, such that and , for every .
Example 1. Here, we include a daily life example of six different countries. As an illustrative case, consider a network (IFG) of six vertices indicating the six different countries. The MS value of the vertices represents the percentage of people who are educated and the NMS value of the vertices indicates the percentage of those people who are uneducated. The MS value of the edges expresses the cooperation of one country with another country to enhance the percentage of educated people and the NMS value denotes the noncooperation with each other.
Figure 1 shows an IFG with andIn this paper, the minimum operator is expressed by and the maximum operator is represented by .
Definition 3 (see [23]). Let be an IFG and ; then, its degree is represented by and defined by and .
Definition 4 (see [26]). Assume is a graph. Then, is named as an incidence graph (IG), where .
Example 2. An IG is shown in Figure 2. are called incidence pairs (IPs) or pairs of an IG .
Definition 5 (see [26]). Assume is a graph, is a FSS of V, and is a FSS of . Let be a FSS of . If for every , then is a fuzzy incidence (FI) of G.
Definition 6 (see [26]). Assume G is a graph and is a fuzzy subgraph of . If is a FI of , then is named as FIG of G.
Example 3. A FIG with and is shown in Figure 3.
3. Degree of a Vertex of CP and Composition of Intuitionistic Fuzzy Incidence Graphs
In this section, we define the IFIG, CP, and composition of two IFIGs. We also define the degree of a vertex of these products on IFIGs. Theorems are also provided to calculate the degree of a vertex in CP and composition.
Definition 7. An IFIG is of the form where , , , and such that and represent the degree of MS and NMS of the vertex , respectively, and for each and ; and show the degree of MS and NMS of the edge , respectively, such that and , for every and ; and show the degree of MS and NMS of the incidence pair, respectively, such that and , for every .
Example 4. Here, we include a daily life example of three different shopping malls. As an illustrative case, consider a network (IFIG) of three vertices indicating the three different branches of a shopping mall. The MS value of the vertices represents the number of people who visit the shopping mall for shopping and the NMS value of the vertices indicates those people who do not come to these shopping malls for shopping. The MS value of the edges expresses their cooperation with each other and the NMS value denotes the noncooperation with each other. The MS value of the IPs shows the percentage of profit and the NMS value indicates the percentage of nonprofit of these shopping malls.
Let be an IFIG provided in Figure 4, expressing a network of three branches of a shopping mall where , , and . Let and , , and
Definition 8. Consider is an IFIG and ; then, its degree is expressed by and defined by and .
Now, we are going to define the CP of two IFIGs and how to find the degree of these graphs.
Definition 9. The CP of two IFIGs and is defined as an IFIG where , or , and or with , .
Definition 10. Let be the CP of two IFIGs and . Then, the degree of is expressed by and defined by
Theorem 1. Let and be two IFIGs. If and , then .
Proof. In CP, by the definition of the degree of a vertex, we haveSince and ,Since and ,Hence, .
Example 5. In Figures 5 and 6,, and . Then, by Theorem 1, we haveSo, . In a similar manner, we can examine the degree of all the vertices of CP of Figures 5 and 6 provided in Figure 7.
Next, the method to explore the degree of a vertex in the composition of two IFIGs is provided.
Definition 11. The composition of two IFIGs and is defined as an IFIG where , or , and or with
Definition 12. Let be the composition of two IFIGs and . Then, the degree of is represented by and defined by
Theorem 2. Let and be two IFIGs. If and , then .
Proof. In the composition of two IFIGs by definition of degree of a vertex, we haveThis implies .
Example 6. The composition of graphs given in Figures 5 and 6 is shown in Figure 8. Here, and ; then, by Theorem 2,